Stochastic Regression
In classical linear regression model, the regressor is
assumed to be fixed. now if this assumption is not met and the response
variable is depended on non-fixed regressor (called stochastic regressor). In
stochastic regression the regressor is selected randomly from a population with
finite mean and finite non - zero variance.
Assumptions
i.
The mean of disturbance term is zero
ii. The variance of the disturbance is constant.
iii.
The disturbance terms for different
variables are independent.
iv.
The regressor is measured with error or
regressor is correlated with error.
v.
The response, predictor and disturbance
term is random.
vi. The disturbance follows normal with mean zero and constant variance.
Estimation by OLS Method
The OLS estimate of model
Properties:
i.
The OLS estimates will be biased and in
case of stochastic regressor.
ii.
The OLS estimates will be inconsistent
The covariance between regressor and error term is not zero due
to stochastic regressor.
Thus,
i.
Independent variable is used as response
variable in another model. (Simultaneous Equations)
ii.
Independent variable is measured with
error. (Error in Variables)
Endogenous Variables
Endogenous variable (or endogenous
variable(s)) is the variable(s) that are produced by an econometric (or
economic) model. Both the system and endogenous variables have an impact on one
another.
Consider the system of
equations given below:
Exogenous Variables
Exogenous
variables are those that come from sources outside the economic (or
econometric) model. Exogenous variables are fixed and presumed to be
independent of the disturbance term. Exogenous variables influence the system
but are not influenced by the system.
In the above
equations: X1 and X2 are exogenous
variables.
Statistically,
exogeneity means that
The lagged value of the
endogenous variable is also fixed, so it is considered as exogenous variable.
Simultaneous
Equations Models
(SEM)
A simultaneous
equation model is a set of simultaneous linear equations, in which the
dependent variable (called endogenous) is a function of an independent
variable(s) (called exogenous) and also the endogenous variable(s) appears as
an exogenous variable(s) in the other equation.
This
indicates that some exogenous variables are simultaneously determined with the exogenous
variable, which is typically the result of an underlying equilibrium mechanism
in economics.
A
simultaneous equation model is given by;
Since there is a joint dependency in a simultaneous equation model, the OLS method cannot be employed to estimate the parameters.
e.g., A simple model
of demand and supply.
In the model, Q and P are
endogenous variables and Income “I” is exogenous variable. The two equilibrium
values, for P and Q determine at the same time.
Further it is assumed
that,
i.
The
quantity and supply models
ii.
Keynesian
income determination model
iii.
Consumption
and tax models
Types of Simultaneous Equations Model
1.
Complete
Simultaneous Equations Models
a simultaneous equation model in which the total number of endogenous variables is equal to the number of equations in the model.
That’s
Number of endogenous variables = Number of equations
e.g.
2.
Incomplete
Simultaneous Equations Models
A simultaneous equation model, in which the total number of endogenous
variables is more than the number of equations.
That’s
Number of endogenous variables > Number of equations
e.g.
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